Finite Group Databasebeta

← #11724125

Subgroups — #11724125

Order 864.

236 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 42 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#88957C212yesmaximal
2#88940C211yesmaximal, characteristic
3#88934C212yesmaximal
4C3×#3773C211yesmaximal, characteristic
5Dic3×#157C211yesmaximal, characteristic
6#64222—31nomaximal
7C2×C3⋊#157C2211yescharacteristic
8C2×C3×#157C2211yescharacteristic
9#6294C2212yes—
10#6293—22no—
11Dic3×C3⋊D3C2211yescharacteristic
12C3×#627—22no—
13C3×#624C2212yes—
14#3797—32no—
15D3×Dic6—62no—
16C4×#157—31no—
17#3773—31no—
18#3773—32no—
19#3773D311yescharacteristic
20C2×C3×C3⋊D3C2311yescharacteristic
21C3⋊#157Q812yescomplemented
22C3×#157Q812yescomplemented
23C3⋊(C3⋊Dic3)D411yescharacteristic
24C3×C3⋊Dic3—22no—
25C32×Dic3—22no—
26C32×Dic3D411yescharacteristic
27#888—91nomaximal
28C2×#157—31no—
29C2×#157C2×D311yescharacteristic
30C4×C3⋊D3—31no—
31#634—62no—
32C3×C4×D3—62no—
33C3×Dic6—62no—
34C3×Dic6—62no—
35#627—22no—
36#627—62no—
37#627—62no—
38#627—122no—
39#624—32no—
40#624C2×D312yes—
41D3×Dic3—62no—
42D3×Dic3—62no—
43C2×C33C2×D411yescharacteristic
44C3×C3⋊D3C2×Q811yescharacteristic
45C3×C3⋊D3C2×Q811yescharacteristic
46D3×Q8—182no—
47C2×Dic6—92no—
48C3×C4⋊C4—91no—
49C4⋊Dic3—91no—
50Dic3⋊C4—92no—
51C4×Dic3—91no—
52C2×C3⋊D3C22×D311yescharacteristic
53C2×C3×D3—62no—
54#157—32no—
55#157Dic612yescomplemented
56C32×C4—31no—
57C32×C4—62no—
58C3⋊Dic3—22no—
59C3⋊Dic3—31no—
60C3⋊Dic3—62no—
61C3⋊Dic3—122no—
62C3×Dic3—22no—
63C3×Dic3—22no—
64C3×Dic3—22no—
65C3×Dic3—42no—
66C3×Dic3—62no—
67C3×Dic3—62no—
68C3×Dic3—122no—
69C4⋊Q8—271noSylow
70C33C4⋊Q811yescharacteristic, complemented, Sylow
71C3×Q8—182no—
72C2×C3×C4—92no—
73C2×C3×C4—91no—
74C2×Dic3—92no—
75C2×Dic3—91no—
76C2×Dic3—91no—
77C4×D3—62no—
78C4×D3—182no—
79C4×D3—182no—
80Dic6—62no—
81Dic6—62no—
82Dic6—182no—
83Dic6—182no—
84Dic6—182no—
85Dic6—362no—
86C2×C32—22no—
87C2×C32—42no—
88C2×C32D3×D411yescharacteristic
89C3⋊D3C2×Dic611yescharacteristic
90C3⋊D3D3×Q811yescharacteristic
91C3×D3—62no—
92C3×D3—62no—
93C2×Q8—272no—
94C4⋊C4—91no—
95C4⋊C4—272no—
96C4⋊C4—271no—
97C42—271no—
98C22×C3—91no—
99C2×D3—62no—
100C3×C4—62no—
101C3×C4—62no—
102C3×C4—62no—
103C3×C4—92no—
104C3×C4—122no—
105C3×C4—182no—
106Dic3—22no—
107Dic3—62no—
108Dic3—62no—
109Dic3—92no—
110Dic3—91no—
111Dic3—122no—
112Dic3—182no—
113Dic3—362no—
114Dic3GO+(4,𝔽2)11yescharacteristic
115C32—22no—
116C32—42no—
117C32#88811yescharacteristic, complemented
118Q8—182no—
119Q8—542no—
120C2×C4—92no—
121C2×C4—91no—
122C2×C4—271no—
123C2×C4—272no—
124C2×C4—271no—
125C2×C3—22no—
126C2×C3—42no—
127C2×C3—91no—
128C2×C3—91no—
129C2×C3C2×GO+(4,𝔽2)11yescharacteristic
130D3—62no—
131D3—62no—
132C22—91no—
133C4—31no—
134C4—62no—
135C4—92no—
136C4—182no—
137C4—272no—
138C4—271no—
139C3—22no—
140C3—42no—
141C3#6422211yescharacteristic, complemented
142C2—91no—
143C2—91no—
144C2D3×GO+(4,𝔽2)11yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2D3×GO+(4,𝔽2)
commutator, prime radical, cosocleC2×C3×C3⋊D3C23
Fitting, socleC2×C33C2×D4
hypocenterC33C4⋊Q8
LCSG ▷ C2×C3×C3⋊D3 ▷ C33
UCSC1 ◁ C2

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
21. C3⋊#157Q8semidirect
22. C3×#157Q8semidirect
55. #157Dic6semidirect
70. C33C4⋊Q8semidirect
117. C32#888semidirect
141. C3#64222semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#889572
#889402
#889342
C3×#37732
Dic3×#1572
#642223
#8889